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according alter abruptly angle angular velocities approximation axes axis become infinite Cambridge Philosophical Cambridge Philosophical Society canal centre co-ordinates consequence considered convergent series corresponding cosines cylinder denote density depends depth determined direction discontinuous displacements distance dx dy elastic solids equal equations of motion ether exact differential expansion expression finite fluid motion formula free surface friction function given h vanishes Hence homogeneous function hypothesis incompressible fluid increases indefinitely inertia integral limit luminiferous ether mass memoir molecules moment of inertia moving neglected observed obtained order of approximation oscillation paper parallel particles perpendicular plane Poisson Poisson's equations positive quantities ratio rectangular refracting refracting medium respect rest result satisfied shew shewn sphere suppose supposition tangential force theorem theory tion variables velocity of propagation waves whence zero
الصفحة 114 - The capability which solids possess of being put into a state of isochronous vibration shews that the pressures called into action by small displacements depend on homogeneous functions of those displacements of one dimension. I shall suppose moreover, according to the general principle of the superposition of small quantities, that the pressures due to different displacements are superimposed, and consequently that the pressures are linear functions of the displacements.
الصفحة 142 - ... in the terrestrial methods of determining the velocity of light, the light comes back along the same path again, so that the velocity of the earth with respect to the ether would alter the time of the double passage by a quantity depending on the square of the ratio of the earth's velocity to that of light, and this is quite too small to be observed.
الصفحة 181 - ... accurate means of comparing theory and experiment in a class of motions in which they have not hitherto been compared, so far as I am aware. When a box of the form of a rectangular parallelepiped, filled with fluid and closed on all sides, is made to perform small oscillations, it appears that the motion of the box will be the same as if the fluid were replaced by a solid having the same mass, centre of gravity, and principal axes as the solidified fluid, but different principal moments of inertia....
الصفحة 126 - there seems no line of demarcation between a solid and a viscous fluid...
الصفحة 281 - U unless the convergency of the series (67) becomes infinitely slow when h vanishes. 'The convergency of the series is here said to become infinitely slow when, if n be the number of terms which must be taken in order to render the sum of the neglected series numerically less than a given quantity e, which may be as small as we please, n increases beyond all limit as h decreases beyond all limit. 'DEMONSTRATION. If the convergency do not become infinitely slow it will be possible to find a number...
الصفحة 182 - It is not absolutely necessary thaty(r") and F (r) should always have the same sign. In forming the equations of motion M. Navier adopts the hypothesis of a symmetrical arrangement of the particles, or at least, which leads to the same result, neglects the irregular part of the mutual action of neighbouring molecules. The equations at which he arrives are those which would be obtained from the common...
الصفحة 165 - Gréent» who has considered the case in which the depth is infinite. The oscillatory waves just considered are those which are propagated uniformly in fluid of which the depth is everywhere the same. When this condition is not satisfied, as for instance when the waves are propagated in a canal whose section is not rectangular, it is desirable to know how the velocity of propagation and the form of the waves are modified by this circumstance. There is one such case in which a solution has been obtained....
الصفحة 158 - Tom. ii. p. 42. III. The discharge of gases through small orifices. IV. Theory of sound. V. Simultaneous oscillations of fluids and solids. VI. Formation of the equations of motion when the pressure is not supposed equal in all directions. I. Although the common equations of hydrodynamics have been so long known, their complexity is so great that little has been done with them except in the case in which the expression usually denoted by udx+vdy+wdz (A.) is the exact differential of a function of...
الصفحة 50 - A similar result may be proved to be true for any solid symmetrical with respect to two planes at right angles to each other, and...
الصفحة 88 - In most cases in which it would be interesting to apply the theory of friction of fluids, the density of the fluid is either constant or may without sensible error be regarded as constant, or else changes slowly with the time. In the first two cases the results would be the same and in the third nearly the same whether T)v were equal to zero or not.